A method of operating a microlithographic projection exposure apparatus and projection exposure apparatus

The invention relates to a method of operating a microlithographic projection exposure apparatus, and a microlithographic projection exposure apparatus, wherein the illumination device is arranged a in an object plane of the projection objective, structures to be imaged mask having (131, 207) illuminated with useful light of an operating wavelength, and wherein the projection objective these structures to an in arranged an image plane of the projection lens substrate (305, 402, 500) mapping, wherein in the substrate (305, 402, 500) at least temporarily a non caused by the useful light heating and / or cooling power is coupled such that without this coupling at the imaging of the structures generated by the projection exposure system optical aberration and / or a coupling without these generated during operation of the projection exposure apparatus deformation of the substrate (305, 402, 500) is at least partially compensated.